A European Informational Website
learn more
In mathematics, multiplication is an elementary arithmetic operation. When one of the numbers is a whole number, multiplication is the repeated sum of the other number.
For example, 7 × 4 (verbally, "seven times four") is the same as 7 + 7 + 7 + 7, or 4 + 4 + 4 + 4 + 4 + 4 + 4 (four sevens or seven fours).
Fractions are multiplied by separately multiplying their denominators and numerators: a/b × c/d = (ac)/(bd). For example, 2/3 × 3/4 = (2×3)/(3×4) = 6/12 = 1/2.
Multiplication can be defined for real and complex numbers, polynomials, matrices and other mathematical quantities as well; see product (mathematics). The inverse of multiplication is division.
For several ways to compute products, including very large numbers, see multiplication algorithms.
The standard methods for multiplying numbers using pencil and paper require a multiplication table of memorized or consulted products of small numbers (typically any two numbers from 0 to 9), however one method, the peasant multiplication algorithm, does not. Many mathematics curricula developed according to the 1989 standards of the NCTM do not teach standard arithmetic methods, instead guiding students to invent their own methods of computation. Though widely adopted by many school districts in nations such as the United States, they have encountered resistance from some parents and mathematicians, and some districts have since abandoned such curricula in favor of traditional mathematics.
Multiplying numbers to more than a couple of decimal places by hand is tedious and error prone. Common logarithms were invented to simplify such calculations. The slide rule allowed numbers to be quickly multiplied to about three places of accuracy. Beginning in the early twentieth century, mechanical calculators, such as the Marchant, automated multiplication of up to 10 digit numbers. Modern electronic computers and calculators have greatly reduced the need for multiplication by hand.
Methods of multiplication were documented in the Egyptian, Greece, Babylonian, Indus valley, and Chinese civilizations.
The Egyptian method of multiplication of integers and fractions, documented in the Ahmes Papyrus, was by successive additions and doubling. For instance, to find the product of 13 and 21 one had to double 21 three times, obtaining 2 × 21 = 42, 4 × 21 = 84, 8 × 21 = 168. The full product could then be found by adding the correct terms found in the doubling: (note 13 = 1 + 4 + 8)
The Babylonians used a sexagesimal positional number system, analogous to the modern day decimal system. Thus, Babylonian multiplication was very similar to modern decimal multiplication. Because of the relative difficulty of remembering 60 × 60 different products, Babylonian mathematicians employed multiplication tables. These tables consisted of a list of the first twenty multiples of a certain principal number n: n, 2n, ..., 20n; followed by the multiples of 10n: 30n 40n, and 50n. Then to compute any sexagesimal product, say 53n, one only needed to add 50n and 3n computed from the table.
In the books, Chou Pei Suan Ching dated prior to 300 B.C., and the Nine Chapters on the Mathematical Art, multiplication calculations were written out in words, although the early Chinese mathematicians employed an abacus in hand calculations involving addition and multiplication.
The early Hindu mathematicians of the Indus valley region used a variety of intuitive tricks to perform multiplication. Most calculations were performed on small slate hand tablets, using chalk tables. One technique was that of lattice multiplication (or gelosia multiplication). Here a table was drawn up with the rows and columns labelled by the multiplicands. Each box of the table was divided diagonally into two, as a triangular lattice. The entries of the table held the partial products, written as decimal numbers. The product could then be formed by summing down the diagonals of the lattice.
The two numbers being multiplied are formally called the multiplicand and the multiplier, respectively. (Some write the multiplier first, and say that 7 × 4 stands for 4 + 4 + 4 + 4 + 4 + 4 + 4, but this usage is less common.) The difference was important in Roman numerals and similar systems where multiplication is transformation of symbols and their addition. For example, to multiply VII by XV one changes the VII to LXX (multiplying VII by X) plus XXV (V times V) plus X (II times V), but to multiply XV by VII one changes XV into LXXV (XV times V) plus XV plus XV (each XV times I).
Because of the commutative property of multiplication, there is generally no need to distinguish between the two numbers so they are more commonly referred to as the factors. The result of the multiplication is referred to as the product.
Multiplication can be denoted in several equivalent ways. All of the following mean, "5 multiplied by 2":
5×2 (see ×) 5·2 (5)2, 5(2), (5)(2), 5[2], [5]2, [5][2] 5*2 5.2
The asterisk (*) is often used on computers because it is a symbol on every keyboard, but it is rarely used when writing math by hand. This usage originated in the FORTRAN programming language.
Frequently, multiplication is implied by juxtaposition rather than shown in a notation. This is standard in algebra, taking forms like
5x and xy
This notation is potentially confusing if variables are permitted to have names longer than one letter, as in computer programming languages. The notation is not used with numbers alone: 52 never means 5 × 2.
If the terms are not written out individually, then the product may be written with an ellipsis to mark out the missing terms, as with other series operations (like sums). Thus, the product of all the natural numbers from 1 to 100 can be written . This can also be written with the ellipsis vertically placed in the middle of the line, as .
Alternatively, a product can be written with the product symbol, which derives from the capital letter Π (Pi) in the Greek alphabet. Unicode position U+220F (∏) is defined a ''n''-ary product for this purpose, distinct from U+03A0 (Π), the letter. This is defined as:
The subscript gives the symbol for a dummy variable ( in our case) and its lower value (); the superscript gives its upper value. So for example:
In case m = n, the value of the product is the same as that of the single factor x<sub>m</sub>. If m > n, the product is the empty product, with the value 1.
One may also consider products of infinitely many terms; these are called infinite products. Notationally, we would replace n above by the infinity symbol (∞). In the reals, the product of such a series is defined as the limit of the product of the first terms, as grows without bound. That is:
One can similarly replace with negative infinity, and
for some integer , provided both limits exist.
The definition of multiplication as repeated addition provides a way to arrive at a set-theoretic interpretation of multiplication of cardinal numbers. In the expression
if the n copies of a are to be combined in disjoint union then clearly they must be made disjoint; an obvious way to do this is to use either a or n as the indexing set for the other. Then, the members of are exactly those of the Cartesian product . The properties of the multiplicative operation as applying to natural numbers then follow trivially from the corresponding properties of the Cartesian product.
For integers, fractions, real and complex numbers, multiplication has certain properties:
x · y = y · x.
(x · y)·z = x·(y · z).
Note from algebra: the parentheses mean that the operations inside the parentheses must be done before anything outside the parentheses is done.
x·(y + z) = x·y + x·z.
1 · x = x.
and this is called the identity property. In this regard the number 1 is known as the multiplicative identity.
This fact is directly received by means of the distributive property: m · 0 = (m · 0) + m − m = (m · 0) + (m · 1) − m = m · (0 + 1) − m = (m · 1) − m = m − m = 0.
So,
m · 0 = 0
no matter what m is (as long as it is finite).
(−1)m = (−1) + (−1) +...+ (−1) = −m
This is an interesting fact that shows that any negative number is just negative one multiplied by a positive number. So multiplication with any integers can be represented by multiplication of whole numbers and (−1)'s.
All that remains is to explicitly define (−1)·(−1):
(−1)·(−1) = −(−1) = 1
However, from a formal viewpoint, multiplication between two negative numbers is (again) directly received by means of the distributive property, e.g:
Other mathematical systems that include a multiplication operation may not have all these properties. For example, multiplication is not, in general, commutative for matrices and quaternions.